Ask for More Than Bayes Optimal: A Theory of Indecisions for Selective Hypothesis Testing

Selective classification is a powerful tool for automated decision-making in high-risk scenarios, allowing classifiers to act only when confident and abstain when uncertainty is high. Given a target accuracy, our goal is to minimize the number of indecisions, which are observations that we do not automate. For difficult problems, the target accuracy may be unattainable without abstaining from making a decision. By using indecisions, we can target a misclassification rate below the Bayes error rate, while minimizing overall indecision mass. We provide a characterization of the optimal risk in selective classification, establishing continuity and monotonicity properties that enable optimal indecision selection. We revisit selective inference via the Neyman-Pearson testing framework, where indecision enables control of Type II error given a fixed Type I error probability. For both classification and testing, we propose a calibration method, and analyze the excess risk of plug-in classifiers and the excess indecision mass produced by accuracy-based calibration. In the binary Gaussian mixture model, we identify an exponent-level phase transition, showing that minimal indecision can yield near-optimal accuracy even under poor class separation. Experiments on Gaussian mixtures and real datasets illustrate how indecision can improve selective accuracy.